MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2002 Swedish Mathematical Competition
2002 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
a tetrahedron has 5 edges of length 3 and circumradius 2
A tetrahedron has five edges of length
3
3
3
and circumradius
2
2
2
. What is the length of the sixth edge?
5
1
Hide problems
a+b =? if a^3 - 3a^2 + 5a - 17 = 0 and b^3 - 3b^2 + 5b + 11 = 0
The reals
a
,
b
a, b
a
,
b
satisfy
{
a
3
−
3
a
2
+
5
a
−
17
=
0
b
3
−
3
b
2
+
5
b
+
11
=
0.
\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}
{
a
3
−
3
a
2
+
5
a
−
17
=
0
b
3
−
3
b
2
+
5
b
+
11
=
0.
Find
a
+
b
a+b
a
+
b
.
4
1
Hide problems
n^{1/(n-7)} an integer for n >=8
For which integers
n
≥
8
n \ge 8
n
≥
8
is
n
1
n
−
7
n^{\frac{1}{n-7}}
n
n
−
7
1
an integer?
3
1
Hide problems
y = ax^2, intersect circle ((0,0),1) besides (0, 0)
C
C
C
is the circle center
(
0
,
1
)
(0,1)
(
0
,
1
)
, radius
1
1
1
.
P
P
P
is the parabola
y
=
a
x
2
y = ax^2
y
=
a
x
2
. They meet at
(
0
,
0
)
(0, 0)
(
0
,
0
)
. For what values of
a
a
a
do they meet at another point or points?
2
1
Hide problems
A, B, C can walk at 5 km/hr, a car with 2 of them travels at 50 km/hr.
A
,
B
,
C
A, B, C
A
,
B
,
C
can walk at
5
5
5
km/hr. They have a car that can accomodate any two of them whch travels at
50
50
50
km/hr. Can they reach a point
62
62
62
km away in less than
3
3
3
hrs?
1
1
Hide problems
268 numbers are written around a circle
268
268
268
numbers are written around a circle. The
17
17
17
th number is
3
3
3
, the
83
83
83
rd is
4
4
4
and the
144
144
144
th is
9
9
9
. The sum of every
20
20
20
consecutive numbers is
72
72
72
. Find the
210
210
210
th number.